3.7.10 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [610]

Optimal. Leaf size=18 \[ -\frac {e^{-3 \coth ^{-1}(a x)}}{3 a c} \]

[Out]

-1/3/a/c*((a*x-1)/(a*x+1))^(3/2)

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6318} \begin {gather*} -\frac {e^{-3 \coth ^{-1}(a x)}}{3 a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)),x]

[Out]

-1/3*1/(a*c*E^(3*ArcCoth[a*x]))

Rule 6318

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac {e^{-3 \coth ^{-1}(a x)}}{3 a c}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} -\frac {e^{-3 \coth ^{-1}(a x)}}{3 a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)),x]

[Out]

-1/3*1/(a*c*E^(3*ArcCoth[a*x]))

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Maple [A]
time = 0.13, size = 24, normalized size = 1.33

method	result	size
gosper	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a c}\)	\(24\)
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{3 a c}\)	\(24\)
trager	\(-\frac {\left (a x -1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{3 a c \left (a x +1\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

-1/3/a/c*((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]
time = 0.26, size = 23, normalized size = 1.28 \begin {gather*} -\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{3 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

-1/3*((a*x - 1)/(a*x + 1))^(3/2)/(a*c)

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Fricas [A]
time = 0.34, size = 34, normalized size = 1.89 \begin {gather*} -\frac {{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{2} c x + a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

-1/3*(a*x - 1)*sqrt((a*x - 1)/(a*x + 1))/(a^2*c*x + a*c)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(-a**2*c*x**2+c),x)

[Out]

-(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**3*x**3 + a**2*x**2 - a*x - 1), x) + Integral(a*x*sqrt(a*x/(a
*x + 1) - 1/(a*x + 1))/(a**3*x**3 + a**2*x**2 - a*x - 1), x))/c

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
time = 0.43, size = 49, normalized size = 2.72 \begin {gather*} \frac {2 \, {\left (3 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 1\right )}}{3 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x + 1\right )}^{3} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

2/3*(3*(a + sqrt(a^2 - 1/x^2))^2*x^2 + 1)/(((a + sqrt(a^2 - 1/x^2))*x + 1)^3*a*c)

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Mupad [B]
time = 1.21, size = 23, normalized size = 1.28 \begin {gather*} -\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3\,a\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a^2*c*x^2),x)

[Out]

-((a*x - 1)/(a*x + 1))^(3/2)/(3*a*c)

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